# Breusch–Pagan (BP) Test Gives Wildly Inaccurate Result

When I conduct a manual BP test in R, like so

model <- lm(Y~X, data = dataset)
summary(model)

# BP Test
adj_model <- lm(resid(model)^2 ~ dataset$X) summary(adj_model)  I get an F statistic of 18.74: > summary(adj_model) Call: lm(formula = resid(model)^2 ~ dataset$X)

Residuals:
Min       1Q   Median       3Q      Max
-186.517  -47.607   -6.497   36.640  269.204

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)  -53.869     36.477  -1.477 0.150889
dataset\$X      8.895      2.055   4.329 0.000173 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 97.41 on 28 degrees of freedom
Multiple R-squared:  0.401,	Adjusted R-squared:  0.3796
F-statistic: 18.74 on 1 and 28 DF,  p-value: 0.0001728


There are 30 observations in the sample, and the model has a multiple R^2 of 0.401 with just the one regressor. So if I use a calculator to get the F statistic, the result is the same as above:

F=\frac{0.401\div(2-1)}{(1-0.401)\div(30-2)}=18.74

But when I use the bptest() function from the lmtest library, I get something completely different:

> bptest(model)

studentized Breusch-Pagan test

data:  model
BP = 12.029, df = 1, p-value = 0.0005237


I thought that this might have something to do with the 'studentisation', but even when I turn this off, I still get a result that isn't close to 18.74:

> bptest(model, studentize = FALSE)

Breusch-Pagan test

data:  model
BP = 12.599, df = 1, p-value = 0.0003859


How can these discrepancies be explained?

I intepret this as asking whether the BT test on your model is identical to calculating the F statistics of your adjusted model. I don't know the answer to that, but perhaps you could ask that to statistic expert(s)?

The standard way to calculate a BP statistic is to use N*(R^2), which is exactly what you're getting. Using the F- as you've done is asymptotically equivalent, but not identical.

Note also that the p-values are effectively the same.

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